Unit IV

The Theory of the Firm

• This chapter is the first of four exploring the theory of the firm.

• A firm (or producer or business) combines inputs — labor, capital, land, and materials — to produce outputs.

  • If the firm is successful, the outputs are more valuable than the inputs.
    
  • Production includes more than just manufacturing — encompassing any process or service that creates value, such as transportation, distribution, and retail.

Key Decisions in Production

• Production involves several important decisions that define a firm’s behavior, including:
  • What product or products should the firm produce?
    
  • How should the firm produce them?
    
  • How much output should the firm produce?
    
  • What price should the firm charge for its products?
    
  • How much labor should the firm employ?

Determinants of Firm Behavior

• The answers to these questions depend on both production and cost conditions and on the market structure of the industry.

• Market structure refers to how competitive an industry is and can be described by questions such as:

  • How much market power does each firm possess?
    
  • How similar are products across firms?
    
  • How difficult is it for new firms to enter the industry?
    
  • Do firms compete primarily on price, advertising, or other product differences?

Spectrum of Competition

Range in Size of U.S. Firms

Source: U.S. Census, 2010

Profit and Cost Basics

• Each business, regardless of size or complexity, tries to earn a profit:
  • \(\pi = \text{Revenue} - \text{Costs}\)
  • \(\text{Revenue} = TR = P \times Q = \text{Price} \times \text{Quantity}\)
  • \(\text{Costs} = TC = FC + VC + OC = \text{Fixed} + \text{Variable} + \text{Opp.}\)

Explicit vs. Implicit Costs

• Explicit costs are out-of-pocket expenses made by the firm.
  • Examples include wages paid to employees or rent for office space.
• Implicit costs represent the opportunity cost of using resources the firm already owns.
  • Examples include working in the business without a formal salary, or using part of a home as retail space.
    
  • Implicit costs also include depreciation of equipment and materials necessary for production.

Fixed and Variable Inputs

• Fixed inputs are those that cannot easily be changed in the short run.
  • For example, abuilding is a fixed input — the owner is bound by the lease until it expires.
• Variable inputs can be increased or decreased easily in the short run.
  • For instance, a pizzeria can order more ingredients or hire another worker with little delay.

Time Horizons

• The short run is the period during which at least one factor of production is fixed.
  • During a lease, the pizza shop operates in the short run because it cannot change its building size.
• The long run is the period during which all factors are variable.
  • Once the lease expires, the owner can relocate to a larger or smaller space.

Accounting vs. Economic Profit

• Accounting profit is a cash-based measure:
  • Total revenue minus explicit costs — the difference between money received and money paid out.
• Economic profit equals total revenue minus total cost, including both explicit and implicit costs.
  • This distinction matters because firms pay taxes (and dividends) on accounting profit, but the efficiency of decisions depends on economic profit.

Accounting vs. Economic Profit Example

• Eryn currently works for a corporate law firm where she makes $125k and is considering opening her own legal practice.
  • She expects to earn $200k per year once she establishes herself.
  • To run her own firm, she would need an office and a law clerk.
  • Office rent: $50k/year; law clerk: $35k/year
• Accounting \(\pi = \text{TR} - \text{EC} = \$200 - (\$50k + \$35k) = \$115k\)
• Economic \(\pi = \text{TR} - \text{EC} - \text{OC} = \$200 - (\$50k + \$35k) - \$125k = -\$10k\)

Production Functions

• Recall our factors of production are: \(Q(N,L,K,E) = f(N,L,K,E)\)

• Often times, we simplify the production function such that \(Q(K,L)\)

• The marginal product of labor (capital) measures how much additional output one more worker (piece of technology) would increases a firm’s output, holding all else constant.

  • \(MP_L = \frac{\partial Q(K,L)}{\partial L} \approx \frac{\Delta Q}{\Delta L}\)

Production Schedule: Example

Production Schedule: Example

Production Schedule: Example

The Law of Diminishing Marginal Product

• As more workers are added, marginal product may increase at first — due to specialization or efficiency gains — but eventually decreases as inputs become crowded or less productive.
  • In some cases, adding more workers may have no effect or even a negative effect on total output.
• This pattern is known as the Law of Diminishing Marginal Product, a key feature of short-run production.
  • The idea parallels diminishing marginal utility in consumer theory.

Cost Relationships

• \(\text{Average Cost} = \dfrac{TC}{Q} = \dfrac{VC + FC}{Q}\)

• \(\text{Marginal Cost} = \dfrac{\Delta TC}{\Delta Q} = \dfrac{\Delta VC + \Delta FC}{\Delta Q} = \dfrac{\Delta VC}{\Delta Q} = \dfrac{VC_2 - VC_1}{Q_2 - Q_1}\)

Cost Curve Example

Cost Schedule Example

z <- data.frame(
  labor = 0:7,
  quantity = c(0, 16, 40, 60, 72, 80, 84, 82),
  fixed_cost = 160,
  variable_cost = 0:7 * 80
)
z$total_cost <- z$fixed_cost + z$variable_cost
z

  labor quantity fixed_cost variable_cost total_cost
1     0        0        160             0        160
2     1       16        160            80        240
3     2       40        160           160        320
4     3       60        160           240        400
5     4       72        160           320        480
6     5       80        160           400        560
7     6       84        160           480        640
8     7       82        160           560        720

Cost Schedule Example

z$average_cost <- z$total_cost / z$quantity
z$marginal_cost <- NA
z$marginal_cost[2:nrow(z)] <- diff(z$total_cost) / diff(z$quantity)
z

  labor quantity fixed_cost variable_cost total_cost average_cost marginal_cost
1     0        0        160             0        160          Inf            NA
2     1       16        160            80        240    15.000000      5.000000
3     2       40        160           160        320     8.000000      3.333333
4     3       60        160           240        400     6.666667      4.000000
5     4       72        160           320        480     6.666667      6.666667
6     5       80        160           400        560     7.000000     10.000000
7     6       84        160           480        640     7.619048     20.000000
8     7       82        160           560        720     8.780488    -40.000000

Cost Curves Example

Cost Schedule Example

Change in cost per worker

z$average_cost <- z$total_cost / z$labor
z$marginal_cost <- NA
z$marginal_cost[2:nrow(z)] <- diff(z$total_cost) / diff(z$labor)
z

  labor quantity fixed_cost variable_cost total_cost average_cost marginal_cost
1     0        0        160             0        160          Inf            NA
2     1       16        160            80        240     240.0000            80
3     2       40        160           160        320     160.0000            80
4     3       60        160           240        400     133.3333            80
5     4       72        160           320        480     120.0000            80
6     5       80        160           400        560     112.0000            80
7     6       84        160           480        640     106.6667            80
8     7       82        160           560        720     102.8571            80

Cost Curves Example

Change in cost per worker

Output Schedule Example

## made up data, workers and a production function
z <- data.frame(l = 1:100)
z$q <- 2*log(z$l + 4) + 2*log(max(z$l)) + 1 - 2*log(max(z$l) + 4)
z <- rbind(data.frame(l = 0, q = 0), z)

# marginal product of labor
z$mp_l <- NA
z$mp_l[2:nrow(z)] <- diff(z$q) / diff(z$l)

head(round(z, 1))

  l   q mp_l
1 0 0.0   NA
2 1 4.1  4.1
3 2 4.5  0.4
4 3 4.8  0.3
5 4 5.1  0.3
6 5 5.3  0.2

Output Curves Example

Output Schedule Example

# costs (variable + fixed)
wage <- 10
z$total_cost <- z$l * wage + 1000
z$average_cost <- z$total_cost / z$q

# Marginal cost via discrete differences
z$marginal_cost <- z$total_cost / z$q
z$marginal_cost[2:nrow(z)] <- diff(z$total_cost) / diff(z$q)

## peak at the data
head(round(z, 1))

  l   q mp_l total_cost average_cost marginal_cost
1 0 0.0   NA       1000          Inf           Inf
2 1 4.1  4.1       1010        243.9           2.4
3 2 4.5  0.4       1020        226.4          27.4
4 3 4.8  0.3       1030        214.0          32.4
5 4 5.1  0.3       1040        204.7          37.4
6 5 5.3  0.2       1050        197.5          42.5

Cost Curves Example #2

Output Schedule Example

Change in cost per worker

# costs (variable + fixed)
wage <- 10
z$total_cost <- z$l * wage + 1000
z$average_cost <- z$total_cost / z$l

# Marginal cost via discrete differences
z$marginal_cost <- z$total_cost / z$l
z$marginal_cost[2:nrow(z)] <- diff(z$total_cost) / diff(z$l)

## peak at the data
head(round(z, 1))

  l   q mp_l total_cost average_cost marginal_cost
1 0 0.0   NA       1000          Inf           Inf
2 1 4.1  4.1       1010       1010.0            10
3 2 4.5  0.4       1020        510.0            10
4 3 4.8  0.3       1030        343.3            10
5 4 5.1  0.3       1040        260.0            10
6 5 5.3  0.2       1050        210.0            10

Cost Curves Example #2

Change in cost per worker

Profit Margin

• \(\text{Average } \pi = \dfrac{\pi}{Q} = \dfrac{TR - TC}{Q} = \dfrac{TR}{Q} - \dfrac{TC}{Q}=...\)

• \(...=\dfrac{TR}{Q} - \dfrac{TC}{Q} = \dfrac{P \times Q}{Q} - \dfrac{AC \times Q}{Q} = P - AC\)

• \(\text{Average } \pi = P - AC\)

• Average profit is also known as the profit margin or profit per unit

  • If \(P > AV\), then the firm is profitable
  • If \(P > MC\), then increasing production will increase profits

Production in the Long Run

• The long run refers to a time horizon in which the firm has no fixed inputs — all factors of production can be adjusted.
  • In contrast, in the short run, capital is typically treated as fixed, while labor is variable.
• The long run is not a specific length of time; it depends on the nature of the firm.
  • For example, if a firm has a one-year lease on its factory, then the long run begins after that lease expires, since the firm can now change its facility.
    
  • In the long run, a firm can build new factories, purchase new machinery, or close existing facilities.

Production in the Long Run

Production in the Long Run

• In planning for the long run, a firm compares alternative production technologies (or processes).
  

• Technology refers to all possible methods of combining inputs to produce outputs — it does not necessarily mean a new invention.

• Firms are assumed to seek the lowest-cost technology capable of producing a desired level of output.

Choice of Production Technology

Choice of Production Technology

Choice of Production Technology